Abstract
It is well known that in the case of constant dielectric permittivity and magnetic permeability, the electric field solving the Maxwell’s equations is also a solution to the wave equation. The converse is also true under certain conditions. Here we study an intermediate situation in which the magnetic permeability is constant and a region with variable dielectric permittivity is surrounded by a region with a constant one, in which the unknown field satisfies the wave equation. In this case, such a field will be the solution of Maxwell’s equation in the whole domain, as long as proper conditions are prescribed on its boundary. We show that an explicit finite-element scheme can be used to solve the resulting Maxwell-wave equation coupling problem in an inexpensive and reliable way. Optimal convergence in natural norms under reasonable assumptions holds for such a scheme, which is certified by numerical exemplification.
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The research of the first author is supported by the Swedish Research Council grant VR 2018-03661.
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Beilina, L., Ruas, V. On the Maxwell-wave equation coupling problem and its explicit finite-element solution. Appl Math 68, 75–98 (2023). https://doi.org/10.21136/AM.2022.0210-21
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DOI: https://doi.org/10.21136/AM.2022.0210-21
Keywords
- constant magnetic permeability
- dielectric permittivity
- explicit scheme
- finite element
- mass lumping
- Maxwell-wave equation